Name: ______Block: ______
PHYSICS 11
5.5 - Applications of Vectors the Across the River Problem
Moving a boat across a river is basically a "relative velocity" problem in two directions. We start with the following assumptions to orient ourselves (note : essentially, we are picking our coordinate system - we can pick it any way we want, we just need to be consistent for the rest of the problem) :
What coordinate system are we using?· The boat speed and river speed are constant (and the river flows East to West (Right to Left) in our example).
· We will start the boat on the South side of the river.
· The "line" between the starting dock and the dock straight across will be our "theta = 0" line.
· We will measure the angle THETA as positive for pointing upstream (to the East of North), and negative for downstream (West of North).
· Let's consider the +X direction to be East and -X direction to be West.
· Let's consider the +Y direction to be North (we don't need -Y here). /
What are the equations for the motion of the boat?
The boat will move sideways in the river based on the river's speed, as well as any X component of the Boat speed. The boat will move across the river based only on the Y component of the Boat speed. To find the time for the boat to get to the other side, the y component of the boat velocity, and the distance across are used. To find the distance along the bank (measured from the point opposite your starting point), use the net velocity sideways, and the time across. The equations are shown below :
The Across the River Problem
Example problem 1
You are rowing a boat across a river, aiming directly for the other side. You are rowing at 3 km per hour but the river is flowing at 4 km per hour. If the river is 200 m wide, how far down-river will you land on the other side?
Example problem 2
An airplane is pointing straight east and flying with airspeed of 300 km/hr. there is a southerly wind of 80 km per hour. What is the actual direction of the flight of the plane and what is its ground speed?
Example problem 3
You are pedaling your bike at 20 km per hour along a road running to the east. There is a northerly wind of 15 km per hour. From which direction does the wind appear to be coming toward your face?
Example problem 4
An airplane is heading in the direction 20 degrees North of East. Its airspeed is 400 km per hour. There is a 120 km per hour wind from the direction 30 degrees south of East. What is the ground-speed and the direction of the flight of the plane?
5.5 Assignments
1. A boat can travel 2.30 m/s in still water. If the boat heads directly across a river with a current of 1.50 m/s:
a. What is the velocity of the boat relative to the shore?
2.75 m/s
b. At what angle compared to straight across is it traveling?
q = 33.1˚
c. How far from its point of origin is the boat after 8.0 s?
22 m
d. At what upstream angle (compared to straight across) must the boat travel in order to the other bank directly opposite its starting point? How fast across the stream is it traveling?
2. You are flying a hang glider at 14 km/h in the northeast direction (45°). The wind is blowing at 4 km/h from due north.
a) What is your airspeed?
b) What angle (direction) are you flying?
c) The wind increases to 14 km/h from the north. Now what is your airspeed and what direction are you flying? If your destination is to the northeast, how would you change your speed or direction so you might make it there? Test your answer using the sim.
Vector Problems (Trig. Solutions)
1. How far East has a person walked if he travels 350 m in a direction 25° E of N?
148 m
2. What would be the resulting displacement if a snail crawls 2.0 m north and then 3.0 m east? What is the snail's direction from the starting point?
3.6 m & 33.7˚ N of E
3. Find the magnitude and direction from the horizontal of a 40.0 N upward force and 17.0 N horizontal force. 43.5 N at 67˚
4. A boat travels east at 13 km/hr when a tide is flowing north at 1.2 m/s. Find the actual velocity and heading of the boat.
3.8 m/s at 18.4˚ N of E
5. A person that swims at 3.2 m/s swims straight across a river with a current of 1.4 m/s. What is the resulting velocity of the swimmer (across and down stream)? At what angle compared to straight across is the swimmer moving?
3.5 m/s at 23.6˚
6. The swimmer above decides to swim into the current at such an angle that he will travel straight across. Find the angle (compared to straight across) at which he would have to swim. Calculate the velocity across the stream. 2.9 m/s at 25.9˚
Vector problems (Component or Sine-Cosine Law Solutions)
1. A seagull flying with an air speed of 10.0 km/h is flying north but suddenly encounters a wind of 5.0 km/h at 20° south of east. What will be the new direction and airspeed of the seagull?
9.5 km/h at 60.5˚ N of E
2. A pilot wishes to reach a city 600.0 km away in a direction of 15° S of W in two hours. (v = 300 km/h at this same direction - this is the resultant vector in the vector diagram!) If there is a wind of 70 km/h blowing at 10° W of S. What must be the heading and air speed of the plane?
heading is 2˚ S of W at an airspeed of 278 km/h
3. A plane that can fly at 250 km/h wishes to reach an airport that has a bearing of 25° W of N from its present location. If there is a 50.0 km/h wind blowing directly to the west what should be the heading of the plane. 14.6˚ W of N (set up vector diagram and then use Sine Law) What will be its ground speed? 267 km/h How long would it take to get to the airport if it were 560 km away? 2.1 hours